A radioactive element has a half life of 200 days. The percentage of original activity remaining after 83 days is ____________. (Nearest integer)
(Given : antilog 0.125 = 1.333, antilog 0.693 = 4.93)

Explanation

To solve the problem of finding the percentage of original activity remaining after 83 days for a radioactive element with a half-life of 200 days, we can use the following steps:

### Step 1: Determine the decay constant (λ)
The decay constant (λ) can be calculated using the half-life (t₁/₂) formula:
\[ \lambda = \frac{\ln(2)}{t_{1/2}} \]
Given that the half-life (t₁/₂) is 200 days:
\[ \lambda = \frac{0.693}{200} = 0.003465 \text{ days}^{-1} \]

### Step 2: Calculate the remaining activity after 83 days
The remaining activity (A) can be calculated using the formula:
\[ A = A_0 e^{-\lambda t} \]
Where:
- \(A_0\) is the initial activity,
- \(t\) is the time elapsed (83 days),
- \(e\) is the base of the natural logarithm.

Substituting the values we have:
\[ A = A_0 e^{-0.003465 \times 83} \]

### Step 3: Calculate the exponent
Calculating the exponent:
\[ -0.003465 \times 83 = -0.288495 \]

### Step 4: Calculate \(e^{-0.288495}\)
Using the approximation \(e^{-x} \approx 10^{-x}\) for small values of \(x\), we can convert:
\[ e^{-0.288495} \approx 10^{-0.288495} \]
Using the antilog values provided, we can find:
\[ \text{antilog}(-0.288495) \approx 10^{-0.288495} \approx 0.125 \]

### Step 5: Calculate the percentage of original activity remaining
Now, we can find the percentage of original activity remaining:
\[ \text{Percentage remaining} = \left( \frac{A}{A_0} \right) \times 100 = 0.125 \times 100 = 12.5\% \]

### Step 6: Round to the nearest integer
Rounding 12.5% to the nearest integer gives us:
\[ \text{Final answer} = 13\% \]

### Final Answer
The percentage of original activity remaining after 83 days is **13%**.

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